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NumGy_LV13.qmd
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---
title: "Numerical Simulation Methods in Geophysics, Part 13: A few more details"
subtitle: "1. MGPY+MGIN"
author: "[email protected]"
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# Recap
1. The Finite Difference (FD) method
* Poisson equation in 1D, look into 2D/3D
* diffusion equation in 1D, time-stepping, (1D wave equation)
1. The Finite Element (FE) method
* Poisson and diffusion equation in 1D
* (complex) Helmholtz equation in 2D for EM problems
* solving EM problems and computational aspects
1. Finite Volume (FV) method for advection problems
## The methods
::: {.callout-tip title="The Finite Difference method"}
approximates the partial derivatives by difference quotients (beware $\Delta x$ and $\Delta a$)
:::
::: {.callout-tip title="The Finite Element method"}
approximates the solution through base functions in integrative sense
:::
::: {.callout-tip title="The Finite Volume method"}
approximates the solution by piecewise constant values and keeps conservation law by fluxes
:::
# Boundary conditions
## Mixed boundary conditions
So far...
* Dirichlet Boundary conditions $u=u_0$
* Neumann Boundary conditions $\pdv{u}{n}=g_B$<br>
for vectorial problems $\vb n\cdot \vb E=0$ or $\vb\curl \vb E=0$
In general mixed, also called Robin (or impedance convective) BC
$$ a u + b \pdv{u}{n} = c $$
## Example DC resistivity with point source
$$\div \sigma \grad u = \div \vb j = I \delta(\vb r - \vb r_s)$$
solution for homogeneous $\sigma$ on surface: $u=\frac{I}{2\pi\sigma} \frac{1}{|\vb r - \vb r_s|}$
E-field $\vb E=-\frac{I}{2\pi\sigma} \frac{\vb r - \vb r_s}{|\vb r - \vb r_s|^3}$
normal direction $\vb E \cdot \vb n=-\frac{u}{|\vb r - \vb r_s|}\cos \phi$ purely geometric
$$\pdv{u}{n} + \frac{\cos\phi}{|\vb r - \vb r_s|}=0$$
## Perfectly matched layers
::: {.columns}
::: {.column}
:::
::: {.column}
$$ \pdv{x} \rightarrow \frac{1}{1+i\sigma/\omega} \pdv{x} $$
$$ x \rightarrow x + \frac{i}{\omega}\int^x \sigma(x')\dd x' $$
:::
:::
## Absorbing boundary conditions
wave equation (e.g. in 2D)
$$ \pdv[2]{u}{t} - v^2 \laplacian u = 0 $$
Fourier transform in $t$ and $y$ (boundary direction) $\Rightarrow \omega, k$
$$ \omega^2 \hat{u} - v^2 \pdv[2]{\hat{u}}{x} + v^2 k^2 \hat{u} = 0 $$
ordinary DE with solution $\hat{u}=\sum a_i e^{\lambda x}$ with $\lambda^2 = k^2 - \omega^2/v^2$
# Modern methods
## Solution in wavenumber domain
Fourier transform of 3D problem into wavenumbers
$$\hat F(k_x, y, z)=\int\limits_{-\infty}^{\infty} F(x,y,z) e^{-\imath k_x x} \dd x$$
partial derivative $\pdv[2]{\hat F}{x}=k_x^2 \pdv[2]{F}{x}$
Poisson equation $\nabla_{3D}^2 u=0$ $\Rightarrow$ Helmholtz equation $\nabla_{3D}^2 \hat u - k_x^2\hat u=0$ $\Rightarrow$ solve many 2D problems & get solution by inverse Fourier transform
## Spectral element method
typically used for global wave phenomena
::: {.columns}
::: {.column}
$u=\sum u_i\phi_i(\pv r)$
$\phi$ Lagrangian polynoms $l_i^N=\prod\limits_k^N \frac{\xi-\xi_k}{\xi_i-\xi_k}$
or Chebychev polynoms
:::
::: {.column}
:::
:::
## Discontinuous Galerkin method
typical for hyperbolic problems
weak form of wave equation with fluxes (FV)
$$M\partial_t q(t) - A^T q(t) = -F(a, q(t))$$
$$\Rightarrow \partial_t q=\vb M^{-1}(A^T q(t) - F(a, q(t)))$$
locally for each element & communication through fluxes (like in FV)
## Infinite Elements
## Meshless modelling
## Meshless divergence operator (Wittke, 2017)
# Error estimation and mesh refinement
* get idea of accuracy of the solution
* refinement of cells with high error (e.g. large gradients)
* comparison between successive refinement solutions
## Error estimation (residual-based)
Poisson problem $-\nabla^2=f$ with bilinear form $a(u,v)=\int \grad u \grad v \dd\Omega$
finite-dimensional function space $V_h$: $a(u_h, v_h)=l(v_h)$
estimate error $e_h$ in bilinear form $a(e_h,v)=a(u, v)-a(u_h, v)$
residual $R=f+\nabla^2 u_h$ leads to $a(e_h,v)=\sum\limits_c\int\limits_{\Omega_c} R v \dd\Omega_c$
## Error estimation (recovery-based)
gradients across element boundaries tend to be discontinuous
compare original (unsmoothed) gradient of the solution with improved
$$(E_h)^2 = \int |M(u_h)-\grad u_h|^2\dd\Omega $$
$M$ obtained by smoothing over patch of elements around each element
## Goal-oriented mesh refinement
primal and dual (adjoint) problem (with receiver as hypothetical source)
inner product of solutions
$$\Phi_{lmn}=\int_{c_n} \vb F^l \vb F^m \dd\Omega_n$$